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I would like to acknowledge that Professors Lin and Toomre
of MIT are also interested in the problem of spiral structure,
and that I have benefited from discussions with them as well
as their students.
Kalnajs 1963, p.13

3.1 Working hypothesis and semi-empirical theory

In hindsight, considering the crucial influence that the
Lin & Shu (1964) paper had on the thinking of astronomers, it is
only regretful that Lin did not decide (with or without me) to
publish even earlier, because he certainly had all the physical
ideas contained in our paper well before 1964.
Shu 2001

While Toomre, Hunter and Kalnajs had already presented their first results in the dynamics of flat galaxies, Lin still kept on thinking over the spiral problem. 63 Astronomers in Princeton had convinced him that, despite Chandrasekhar's criticism of Lindblad's theories, 64 the idea itself of a long-lived, shape-preserving spiral pattern is consistent with Hubble's classification system that relates spiral features with a galaxy's morphological type, its steady characteristic, thus suggesting that the spirals are steady as well. This view reminded Lin of wave modes in fluid flows that he had been studying for years back. 65 On purely heuristic grounds, discrete spiral modes seemed to him very reasonable as the natural result of wave evolution, and, if so, the patterns released might be associated with slowly growing or neutral modes. Lin raised this premise to the rank of working hypothesis, and around it as the nucleus he set to develop a semi-empirical theory. 66 It was seen to follow best the "urgent assignment from the astronomers [...] to make some specific calculations" and "to demonstrate the possibility of the existence of quasi-stationary spiral modes from the theoretical point of view [...] with understanding of the dynamical mechanisms relegated to a secondary and even tertiary position" (Lin). 67,68

"The conclusion in the working hypothesis is not proved or deuce, but supported by an accumulation of theoretical analysis and empirical data. The adoption of this working hypothesis is a very important step in the development of a theory of spiral structure. It means that the authors are committed to back it up with the comparison of subsequent predictions with observational data." (Lin)

The coauthor to share Lin's fame and commitment was his student Frank Shu (Shu 1964) 69 who "found it remarkable that a scientist trained as a professional mathematician would place higher priority on empirical facts than deductive reasoning" and believed that "it was this broad-mindedness and clear vision that gave Lin a considerable advantage over his many competitors of the period" (Shu). 70 The Lin and Shu paper "On the spiral structure of disk galaxies" (Lin & Shu 1964, hereinafter LS64), in which "they first demonstrated the plausibility of a purely gravitational theory for density waves by a continuum treatment" (Lin & Shu 1966, p.459), appeared in August 1964. 71

The paper considered small non-axisymmetric disturbances to a razor-thin cold disk and found for them, through the governing hydrodynamic and Poisson equations, wave-like solutions of the type

Equation 12 (12)

each specified by its eigenfunction varphi(r) and a pair of eigenvalues omega and m. For further advancement, the WKBJ-method was applied. It is valid for the case of phase S(r) varying with radius much faster than amplitude A(r), which features the tightly wrapped spirals, ones of small pitch angle between the circumferential tangent and the tangent to the constant-phase line

Equation 13 (13)

Depending on the sign of a radial-wavenumber function k(r) = - partialS / partialr, the spirals are trailing (k > 0) or leading (k < 0) (Fig.8). With A(r) expanded in a series over a small parameter tani = m / kr (i being the pitch angle), the problem is solved to the lowest, i-independent order neglecting the azimuthal force component of spiral gravity. In this case, both leading and trailing arms act as just rings, so that the ensuing dispersion relation

Equation 14 (14)

substantially repeats Toomre's equation (5) for radial oscillations. Importantly, relation (14) is valid for Re{nu2} leq 1. This restricts the radial span of the WKBJ solutions, and in the neutral case Im{nu}= 0 they gain the territory between the Lindblad resonances determined by Eqn (11) and equating the angular speed of an m -armed spiral pattern to a combination

Equation 15 (15)

with the minus/plus sign discriminating, respectively, between the ILR and OLR. The two-armed spirals thus seem preferred as best covering an entire disk (Fig.9).

Figure 8

Figure 8. The WKBJ approximation and the tightly wrapped spiral waves. kr ident k and ktheta << k are the components of the local wavenumber k. lambda = 2pi / k determines the radial interarm spacing; it is small compared to the galactocentric distance r since kr >> 1 (which is equivalent to small pitch angles i << 1).

Such was the mathematical basis of the original Lin-Shu density-wave theory, called elementary by its authors any later (e.g. Bertin & Lin 1996, p.229). It treated wave quantities Omegap, gamma, and m as free parameters burdened with no dynamical imposition, which made the theory so comfortable in imitating spiral grand designs by means of the curves r(theta) given by

Equation 16 (16)

and obtained through the integration of expressions (13) and (14). Sure, the results of this procedure were controvertible, already because the fast-growing waves - exactly those examined in LS64 - ruled out the proclaimed quasi-stationarity. 72 But the authors hoped that random motions, excluded from their analysis, would in fact stave off disk instability as definitively as to impose a state of near-stability open for slowly growing modes until a small but finite amplitude.

Figure 9

Figure 9. The Lindblad resonances as confining the region accessible for the tightly wrapped spiral waves. (a) - a rotation curve for a galaxy disk and its corresponding corotation and m = 2, 4 Lindblad resonances; (b) - the co-scaled view of the two and four armed tightly wrapped spirals.

Toomre (1964) had reflected already on such a state of Q cong 1 as settling once all over the disk-like stellar Galaxy, but yet he found it stable still, at least in our solar region. As a counterpoise, Lin with Shu diagnosed instability for another region, at about r0 = 4 - 5 kpc from the center. With that, they pictured "a galactic disk, which is in part stable and in part unstable" and suggested "the possibility of a balance resulting in a neutral density wave extending over the whole disk and having a scale of the order of (but smaller than) the distance between the stable and unstable regions" (LS64, p. 651). It was this "suggestion of the possibility" that summarized Lin's early reflections and made his basic working hypothesis originally sound as a statement that

"the total stellar population, which has various degrees of velocity dispersion, forms a quasi-stationary spiral structure in space of the general nature discussed above" (LS64, p.651).

As we can see, this statement hinges almost entirely on the opinion that, for our galactic disk to be equally stable at that r0, the velocity dispersion must there exceed cr, min cong 80 ± 10 km/s, which cannot be the case, else "a considerable number of stars with high radial velocities would reach our neighborhood from the interior part of the Galaxy, contrary to observational evidence" (LS64, p.651). But was this opinion (the authors never repeated it) strong enough? First, it meant an inconceivable situation when some massive portion of a stellar galaxy remains unstable during all the period of formation in it of a global quasi-steady pattern. Secondly, and most important for astronomers, it had - already in 1964 - grave objections to the fact that the largest epicyclic deflection of the Lin-Shu "high radial velocity stars" from their `home' radius r0 = 4 - 5 kpc, equaled to Deltar cong r0 cr / V0 21/2, was in frames of Schmidt's model (cited in LS64) 1 - 1.5 kpc only - too little to let those stars even come close from r0, if not reach us. We find that the original QSSS hypothesis of Lin and Shu, called nowadays "a preliminary formulation" only (Bertin & Lin 1996, p.80), rested on a rather weak basis, both dynamical and empirical.

Very interesting in LS64 is the authors' notice on what had made their work get to print so urgently. A passage following their opening discussion of "at least two possible types of spiral theories", one of which "is to associate every spiral arm with a given body of matter" and the other "is to regard the spiral structure as a [quasi-steady] wave pattern", reads:

"Toomre tends to favor the first of the possibilities described above. In his point of view, the material clumping is periodically destroyed by differential rotation and regenerated by gravitational instability. 73 [...] The present authors favor the second point of view [...] Since A. Toomre's (1964) point of view has been published, it seems desirable to publish our point of view even though the work is not yet as complete as the present writers would wish to have it." (LS64, p.646)

This puzzles. Although it is true that from about 1962 onward Toomre suspected - much as Lynden-Bell had already done in his thesis two years earlier, as it turned out - that at least the more ragged-looking spiral structures result primarily from recurrent gravitational instabilities in the plainly dissipative gas layer of a galaxy (Toomre), there was no explicit discussion of any such suspicions in T64 as actually published. One cannot help but think that this accentuated mention of `Toomre (1964)' was more than just a mistaken reference, that actually it betrayed the influence that at least the cited paper had on Lin.

Shu: "Here, I can only speculate, because certainly my foresight then was not as sharply developed as Lin's. Nor was I privy to the developing estrangement between him and Alar Toomre. [...] Lin had been thinking about the problem of spiral structure nonstop since the Princeton conference in 1961. But he had a world-renowned reputation to protect and therefore was loathe to publish anything hasty before he had worked out his ideas mathematically to his satisfaction. [...] Lin (and later, I) felt strongly that spiral structure was, in essence, a normal mode. But by all the standards of what was then known, a normal mode could not be spiral (unless it grew ridiculously fast). Nevertheless, Lin felt sure that one should not do the naive thing of superimposing equal trailing and leading parts when the wave frequency is (nearly) real. And he probably wanted to discover the reason why before publishing anything. Alar's 1964 paper triggered him into premature action". (Shu)

Lin: "The urgency in my submittal of our paper was to present a different perspective, not to fight for priority". "After reviewing the paper again, I think I could not have done much better or even any better". (Lin)

One way or another, we see that by 1964 Lin indeed had had several thoughts and feelings about spiral modes, and he was eager about gaining power to his perspective. At that, he knew of a growing optimism with shearing or evolving density waves 74 and, as well, of the parallel wave-mode interest at Harvard. The T64 paper 75 , apart from its engagements on disk stability, did mention Kalnajs' advancing efforts and, still more glaringly, it also mentioned and already discussed Lin's yet unpublished solutions. 76 This must have put Lin in a position to urgently patent his views, albeit makeshift in argument for want of better mathematics, and in so doing he rather awkwardly exhibited the opponents' preoccupations as an alternative already placed on record.

63 Lin's basic themes still were in hydrodynamics (e.g., Benney & Lin 1962; Reid & Lin 1963). Back.

64 That criticism (Chandrasekhar 1942) concerned only the asymptotic-spiral theory, and it was itself not flawless as attached to confusing empirical data of the 1920's - 30's. Back.

65 "I have been thinking of modes ever since I learned about the fine points of the Hubble classification". (Lin) Back.

66 "I adopted the empirical approach because of my close contacts with the observers (and with Lo Woltjer). Now that I have thought over the situation some more, I think I should admit that it is probably true that my past long-standing experience in the studies of hydrodynamic instability did (as you hinted) play a role in my thinking (although I was not conscious of it). But more important, I also feel (upon reflection) that the reason I adopted the empirical approach is really the natural consequence of my past education. My undergraduate education was in physics (at Tsinghua University of China, where all the major professors in Physics had doctorate degrees from English speaking universities such as Harvard, Caltech, Chicago and Cambridge), with all the pleasant memories of doing the experiments with precision and the satisfaction of having the data checked against theory. My graduate education was primarily at Caltech where I studied under Theodore von Karman. It is also there that I took a course from Fritz Zwicky who first identified the regular spiral structure in the Population II objects of the Whirlpool M51". (Lin) Back.

67 "Despite of my decades of experience with instability of shear flows, I did not bring these matters into the presentation of the 1964 paper, but commented only vaguely about instability. [...] There was no shortage of theoretical astronomers who understood the mechanisms perhaps better than I did; e.g. Lo Woltjer and Donald Lynden-Bell and perhaps even Peter Goldreich (even at that point). Goldreich turned out be the most successful leader in the understanding of the density waves in the context of planetary rings". (Lin) Back.

68 "In hindsight, I think Lin's judgment was accurate considering how quick people were to attack his point of view with proofs of `antispiral theorems' and the like shortly after the publication of LS64". (Shu) Back.

69 "All the original ideas were C.C. Lin's, and my original contributions were mainly to check the equations that he wrote down and posed as problems. (I did find a way to derive the asymptotic relation between density and potential by attacking the Poisson integral directly, but even there I initially blundered in not realizing the necessity of an absolute value on the radial wavenumber. The final derivation presented in the appendix of LS64 is due to Lin). I did considerable reading, however, on the astronomical side and may have contributed some ideas concerning how OB stars form and die in spiral arms. (This was the beginning of my lifelong interest in star formation.) Lin was indeed quite generous to include me as a coauthor on LS64, and I will always be grateful for his guidance and support of a young (I was 19 at the time) undergraduate student". (Shu) Back.

70 "Lin undoubtedly encouraged many of his younger colleagues - like Alar Toomre - to think about the problem of spiral structure. I can only imagine that Lin's treatment of people then much more junior than himself was equally as generous as his treatment of myself. Certainly, he must have discussed with Alar Toomre (and later Chris Hunter) his ideas about this problem. Toomre's early papers on the subject acknowledge this debt of introduction and inspiration. Why then did those early papers not carry Lin's name as a coauthor? I do not know, nor would I dare to probe (by asking either Lin or Toomre) for fear of opening old wounds that are best left closed". (Shu)

One way or another, no alliance was formed between Lin and Toomre. They "diverged in emphasis from the very beginning", so that "there were discussions, but no real collaboration" (Lin). As in agreement with this Toomre recalls that back again at MIT in spring 1963 he did decline Lin's "astonishing suggestion to write some such paper jointly, since he himself had contributed almost nothing very concretely to my gravitational (in)stability insights, and yet also since I likewise felt I had added next to nothing to his own spiral-wave hopes" (Toomre). Back.

71 That the historical Lin & Shu article was referred to as `Lin's (1963) preprint' by Layzer (1964) and as `Lin (1964)' by Toomre (1964) and Kalnajs (1965) as it was about to appear in the fall of 1964 speaks of its urgently extended coauthorship as Lin's last moment decision (so striking for a well-motivated and ambitious scientist).

Anyway, the Lindblad (1964) paper, also considering quasi-stationary circulation and the resulting spirals in differentially rotating galaxies, appeared half a year prior to Lin's patent. The authors had neither contacts nor fresh news on each other's most parallel work, and hardly could have it. "There was no justification to trouble B. Lindblad with a novice being converted, Lin explains. I was waiting for a definitive new prediction before writing to him. Even then I would have done it through P.O. Lindblad for several obvious reasons. Unfortunately, by the time our result came out (IAU Symposium No 31) [see Sect.3.2] he already passed away" (Lin). Even less probable was any contact-making step from the other side. "About that time [fall of 1964] my father was on a trip around the world caused by the inauguration of the Parkes telescope in Australia, P.O. Lindblad recalls. On his way home he passed through the US [...] but he brought no news about density wave theories. [...] I think my father was aware of the existence of the LS64 paper but had not had the time to penetrate it. I know that he was happy to learn from Whitney Shane, who visited us around the beginning of June 1965, that his work on spiral structure had been more and more appreciated recently". (P.O. Lindblad) Back.

72 To soundly fit the empirical 2-3 kpc local-arm spacing in the Milky Way, LS64 chose a combination of angular speed Omegap = 10 km/s/kpc and growth rate gamma = 50 km/s/kpc (!) for their tentative two-armed spiral. Back.

73 "The prevalent thinking among the other prominent theorists of the time - and this included Alar Toomre - was that spiral structure was a chaotic and regenerative phenomenon - `shearing bits and pieces', as Alar later put it in one of his papers". (Shu) Back.

74 Goldreich and Lynden-Bell in England and Julian and Toomre at MIT set to work on this by 1964. Back.

75 The revised version of T64 was submitted in January 1964. Back.

76 Toomre concluded that "whatever differences there may exist between the shorter axisymmetric and non-axisymmetric disturbances, these must in essence be due only to the circumstance of differential rotation" (T64, p.1223). In Lin's hands, in contrast, this `circumstance' still allowed the dispersion relation (14) for non-axisymmetric waves to be rather close to its axisymmetric analog (5), although the waves stood as steady-mode solutions of the WKBJ type. Yet, as well, the governing equations admitted an "altogether different family of approximate non-axisymmetric solutions" (T64, p.1223), with the radial wavenumber proportional to the disk shear rate A(Oort's constant), and growing with time, kr propto At. This meant that a spiral disturbance of the leading form (t < 0) unwrapped, started trailing, and then wrapped tighter and tighter (t > 0). Thus the point was that, on the one hand, differential rotation continuously deforms even the tightly-wound spiral waves of this sort, whereas, on the other hand, these "should probably be regarded as particular superpositions of Lin's solutions" (T64, p.1223). This discordance was thought to be removed by a fuller analysis beyond the WKBJ-limit. Back.

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